A unifying property for distribution-sensitive priority queues

We present a priority queue that supports the operations: insert in worst-case constant time, and delete, delete-min, find-min and decrease-key on an element x in worst-case O(lg(min{w_x, q_x} + 2)) time, where w_x (respectively, q_x) is the number of elements that were accessed after (respectively, before) the last access of x and are still in the priority queue at the time when the corresponding operation is performed. Our priority queue then has both the working-set and the queueish properties; and, more strongly, it satisfies these properties in the worst-case sense. We also argue that these bounds are the best possible with respect to the considered measures. Moreover, we modify our priority queue to satisfy a new unifying property - the time-finger property - which encapsulates both the working-set and the queueish properties. In addition, we prove that the working-set bound is asymptotically equivalent to the unified bound (which is the minimum per operation among the static-finger, static-optimality, and working-set bounds). This latter result is of tremendous interest by itself as it had gone unnoticed since the introduction of such bounds by Sleater and Tarjan [10]. Together, these results indicate that our priority queue also satisfies the static-finger, the static-optimality and the unified bounds.